3.3.3. Roofs, Faulting and Floors

An independent tetradic feature within a (church) building is found in the faulting and the shape of the roof. A classification would follow the same lines as the graphic expressions of the ground plan. The circle, the cross, the square and the polygonal could be seen as the geometrical representatives of the various quadrants (see also fig. 242). The three-dimensional nature of many roofs might add other possibilities to this classification, but this line will not be followed at this point.

Fig. 284 – The four-fold of the supporting pillars culminates in the twelve fold ribs and openings in the cupola of the cathedral in Zamora (Leon, Spain). The church was built between 1151 and 1174, and features a dome, which tries to capture the invisibility of heaven.

The spatial constraints of a ceiling make it an excellent playground for tetradic patterning (or other geometric configurations). The great majority of ceilings has a square or rectangular form in common. The circular dome-shaped has a smaller application, either to express a sense of holiness to the structure or – in sharp contrast – to facilitate supervision. The dome aims at the cosmic dimensions of the First Quadrant (I) and its invisible invisibility (fig. 284). However, the structure is also used in confined places, like prisons and asylums, where the boundaries of the Third Quadrant (III) become overwhelming (see Ch. 3.6.). This quality of the dome leaves an observer in confusion, if this characteristic entity is positioned in a lower division environment.

The circle (I) and square (III) are accompanied by two more roof forms, i.e. the cross (II) and the octagonal/polygonal setting (IV). The cross creates the first four-division of space. An unlimited area is divided from a certain point to the outer directions (fig. 285). The octagonal find its origin in the circle and the space is limited by this very geometric figure (fig. 286).

The curved and/or inverted types of (church) roofs of the Baroque period (1600 – 1750) are forms, which have been developed from the circle. The ellipsoid domes pushed the spatial theme of cyclic movement to its extreme. The St. Francis of the Order of the Knights of the Cross in Prague (Czechia, 1679 – 1688), the Holy Trinity in Bratislava (Slovakia, 1717 – 1745), the Peterskirche in Vienna (Austria, 1702 – 1733) and the Weltenburg Abbey in Kelheim (Germany, 1716 – 1721) are some of the great churches of this period. They exhibit a daring architectonic adventure, with staggering vistas. It is unfortunate that the development – in the light of the European (church) architecture as-a-whole – turned out to be infertile in the end.

Fig. 285 – This vaulting in the Saint Pierre cathedral of Beauvais (France) is primary based on a cross-design and gives a visual impression of widening of space. This type of geometric patterning was used on a large scale, often in the smaller chapels or halls of the church buildings.

Fig. 286 – The vault of the portico on the south side of the Cathedral of Sainte-Cécile in Albi (France) bears the full scale of cross- and octagonal features within an opening and enclosure of space, defying any singular classification (Photo (2009) and drawing by Marten Kuilman).

David STEPHENSON (2005) produced an excellent photo-book on domes in the European architecture, including the before-mentioned elliptical domes and many more. The subtitle of the book speaks of ‘Visions of Heaven’ and the introduction of his photographic work (by Keith Davis) relates the dome with the human attempt to reach the sublime.

This concept was developed in the beginning of the eighteenth century by, amongst others, Alexander Baumgarten (1714 – 1762) and Edmund Burke (1729 – 1797). The devotion to the sublime can be interpreted as the first rationalization of the notion of ‘feeling’, the observational mode of the Fourth Quadrant. The imagination of infinity, as the destination of opposition, leads in this early stage to a sort of ‘delightful horror’ (Burke). The dome as an architectural entity fit perfectly well into an experience of an infinite universe of which the human soul is a small, but integrate part.

The roots of the dome as an architectural feature can be found in the Middle East, in particularly in countries like Iran and Syria. The most elementary construction of the dome shaped roof construction, is known as the Gwirgwini, and found in the Georgia region of the Caucasus Mountains (fig. 287). KHATCHATRIAN (1971) used the term hazarašen for this form. The same solution to build a dome out of stacked material can be seen in the ancient monk shelters on the west coast of Ireland, where they were called beehives.

Fig. 287 – The most elementary dome-shaped ceiling is constructed with piled woods in a circular way. This so-called ‘Gwirgwini’ is known from the Georgian region in the Caucasus.

The use of the dome to cover jails and asylums exposes a dual reality that the same architectural structure can lead to complete different intentions. The dome can become a symbol of restriction and offers a solution for a totalitarian way of observation. Nobody can escape visibility within the state of universal presence created by the dome. In the correctional buildings covered by a dome is nothing left of the unlimited freedom of the universe, which is otherwise suggested in the (religious) symbolism of the dome.

The story of the dome and the ceiling-in-general is, for this very reason, a complicated one and depends foremost of the position of the observer (in a quadralectic communication). The dome points in the direction of heavens and universality, but at the same time to the totalitarian setting of enclosure and limitation. The square or rectangular ceiling could be the result of a simple box-like outlay of architectural space, but it is also able to transfer the unspoken message of the ‘Four Corners of the Earth’.

One of the better-known ceilings from Antiquity was found in the Temple of Hathor at Dendara (Egypt), situated some sixty kilometers north of Luxor. The ceiling was part of the rooftop sanctuary above the ‘Hall of Offerings’ and shows a circular (Egyptian) zodiac upheld by four goddesses and four pairs of helpers. The discovery during the scientific investigations in the aftermath (1820) of the campaign of Napoleon in Egypt (1798 – 1801) and its subsequent transfer to the Louvre Museum in Paris added to its fame.

The building of the temple of Hathor, the goddess of Joy, started under the leadership of Ptolemeas XII Neos Dionysos (125 BC) and continued up to the Roman Emperor Tiberius (60 AD). The temple is a specimen of ’(late) classical’ Egyptian temple architecture, similar to the ones in Edfoe and Esne, with a major four-partioning and a number of smaller rooms as their main characteristic. The entrance is formed by the Hypostyle Hall (4) with eighteen columns headed by Hathor is followed by the Hall of the Feast of Appearance (3). The Hall of Offerings (with two staircases leading to the roof) and the Hall of the Ennead (2) enclose the most Holy Place of the True Victory (1). The emperors Domitian (51 – 96 AD) and Trajan (c. 53 – 117) added the monumental gateways to the complex.

The historical position of the temple is – from a quadralectic perspective – ‘outside’ the actual Egyptian cultural period (see fig. 58) and must be regarded as a Greco-Roman curiosum, belonging to the second half of the Third Quadrant of the Roman cultural period (see p. 126; fig. 88). The enthusiasm of the early nineteenth century researches for the spoils of war might be a sign of historic resonance. The term refers to a similar position of consciousness on the CF-graph. This phenomenon will be again encountered on the pages 470 and 619/620 of this book.

A ceiling with a zodiac can also be found in a hunting lodge near Qusayr ‘Amra in Jordan (fig. 288). The complex consists of a fortress and bath and dates from the mid-eight century (730 – 740). An Arab prince, living during the rule of the Umayyad (661 – 750), initiated this superb place in the desert and had the leisure buildings decorated with scenes from the country life and the hunt, along with musicians, dancing girls and naked bathing woman (FOWDEN, 2004). The zodiac was probably part of an intellectual display of available knowledge and power, which also included the depiction of the then world leaders elsewhere in the building.

Fig. 288 – An outside view of the Qusayr ‘Amra complex in Jordan with the dome of the tepidarium (warm room). A zodiac is painted inside the dome, indicating the then state of knowledge of the universe, and a consciousness of time and place.

Lester J. NESS (1990; 2000) wrote an informative dissertation on the relation between astrology and Judaism in the Near East and pointed to the occurrence of zodiac mosaics in synagogues in Israel and at the ceiling of temples, like the Temple of Bel in Palmyra (Syria).

Reputedly, the widest single-span vault of unreinforced brickwork in the world was constructed in the Palace of Chosroes I, the Persian (Sassanid) ruler from 531 – 579. He extended the Persian power to the east (Bactria, 560) and to the west (Yemen) and fought against the Romans. His grand-son Chosroes II (588 – 628?) had a less expansional track record, because he first had to deal with the usurper Bahram Chubin. Chosroes II became known during his long reign for his luxury and sumptuousness. Patronage of the arts and philosophers was another of his characteristics.

Chosroes II figured in the legend of the True Cross, which refers to the wood of Christ’s cross, earlier stolen from the Church of the Holy Sepulchre in Jerusalem. There was a strong awareness of heaven. Historians described Chosroes’ II throne at Gandjak (in present-day Azerbaijan) as having a gold and lapis lazuli baldachin with symbols of the sky, the stars and the signs of the zodiac.

The Byzantine ruler Heraclius (610 – 641) recovered the cross (Restitutio crucis) in Persia. The scene with Chosroes, sitting in the astrological tower in the city of Ctesiphon on a mechanical throne (kept in motion by horses, constantly moving like the universe) became an iconological motif and resulted in the feast of the ‘Exultation of the Cross’ at the fourteenth of September (BAERT, 2005). Chosroes II was defeated by Heraclius I and murdered by his son and successor Kavadh II Shiruya.

A special place of interest is the ceilings in Palmyra, the Syrian desert city with such an energetic past. The Zodiac ceiling in the Temple of Bel (fig. 289) and other ceilings (fig. 290) reflect an advanced numerology, which was present at the time (first centuries AD). The circular and lozenge were prominent, but the honeycomb pattern in the inner circle (six-fold) and the zodiac itself (twelve-fold) figured as well. Such a variety of divisions often indicate a ‘decadent’ stage, when the structural constraints of a mental frame are abandoned and ‘anything will do’ as long as the multitude is served.

Fig. 289 – The zodiac ceiling in the Temple of Bel in Palmyra provides a representative specimen of a multiple approach to decoration.

Fig. 290 – The geometric patterns on a ceiling in Palmyra (Syria) are seen here as reproduction’s of Robert Wood’s book on ‘The Ruins of Palmyra. The book was published in London in 1753. It inspired many followers looking for ‘classical’ motives in the revivals that followed.

The symmetrical components comprised here the four-fold and the hexagonal. The geometrical aspect and multitude seemed to be more important than the actual number of division.

The Eastern Mediterranean guidelines for the construction of ceilings found their way into southern Europe with the Muslim expansion of the seventh and eight century. The Moorish style in Spain provides many examples of geometric vaulting along tetradic lines. The Mosque Bib al-Mardum in Toledo (Spain), which was completed in 999, embodied in its ceilings a spectacular display of architecture (fig. 291 – 292).

Fig. 291 – The central vault of the Mosque Bib al-Mardum in Toledo (Spain) is a vigorous graphic statement with a double octagonal design within a square.

Islamic architecture has always been characterized by a strong geometric inclination towards the four-fold. The tetradic design was part of its creative and innovative apparatus. The Muslim domination in Spain lasted from AD 711 up to the Reconquista. The latter movement started with the capture of Toledo in 1085 and ended in 1492, when the Kingdom of Granada surrendered to Ferdinand V and Isabella I. The Bib al-Mardum Mosque in Toledo, the Great Mosque (Mezquita) in Cordoba, and the Alhambra in Granada are examples of the finest (tetradic) architecture ever seen. It will be no surprise, that the Salla de los Abencerrajes (c. 1333 – 1391) in the Alhambra was inspired by the Qur’an’s images of paradise.

One of the great enigmas of Islam – and, to a lesser degree of Christianity – is the combination of its violent expansion and the intention to experience peace and beauty – as preserved in the fourfold. The answer to this issue is of existential importance, because it brings the choice of division-thinking into the middle of a debate.

Fig. 292 – The four- and eightfold division is seen here in some of the smaller domes in the mosque Bib al-Mardum in Toledo (Spain). The building was completed in AD 999.

The Roman tradition of ceiling decoration found an excellent expression and preservation in the ash-buried buildings of Pompeii, where the eruption of the Vesuvius in 79 AD so miraculous arrested a city in time. The ceiling in the House of the Lovers is painted in a tetradic way and dated from the third quarter of the first century BC (fig. 293).

Excellent examples of decorated vaults can be found in Rome – in a house buried beneath the Baths of Caracalla, dating from the Hadrianic period, 130 – 140 AD. Ostia had the ‘Insula of the painted Vaults’, an octagonal decoration from early third century AD (LING, 1991).

Fig. 293 – The restored decoration of the ceiling in the House of the Lovers (Casa degli Amanti) in Pompeii dated from the third quarter of the first century.

The houses of the rich families in the Roman Empire carried on a tradition of interior design, which continues to the present day. The intricate treatment of space and division, like it was developed in the House of Menander (Casa del Menandro) in Pompeii (LING, 2005), is still a guideline in modern concepts. The Roman architect Vitruvius, living in the first century BC, distinguished four styles in the type of atrium, the kernel of the Roman house, depending on the way the rainwater was collected (by means of the compluvium). 1. The Atrium Tuscanicum was formed by two pairs of beams crossing each other at right angles (fig. 294); 2. The Atrium Tetrastylon when the beams were supported at their intersections by pillars or columns; 3. The Atrium Corinthium, which had more than four supporting pillars; 4. The Atrium displuviatum, which sloped towards the outer walls and water was carried off by gutters (JOHNSTON, 1903/1932).

Fig. 294 – The complivium (roof) in a Tuscan atrium leads the rainwater towards the centre, where the water was led into a basin. A curtain could be drawn over the opening when the heat of the sun was too intense.

A further highlight of the decoration of ceilings can be found in the Roman catacombs. Herbert STÜTZER (1983) gave several examples, which have their tetradic layout in common. The underground system of corridors and rooms named after Peter and Marcellius (early fourth century) might be seen as the pinnacle of catacomb development. The symbolism of the walls and ceilings include Christ as the Good Shepherd.

A distinct tetragonal motif is displayed in the Zeno chapel of the Santa Prassede church in Rome. The funerary chapel was built by Pope Paschal I between 817 and 824 for his mother Theodora and contains some beautiful mosaics. Christ as Pancreator supported by four angels offers a heavenly view of the four-fold division (fig. 295). The chapel, and its tetradic design, must have reflected a general idea of higher division thinking, even this mental setting could not be understood and/or described at that time.

Fig. 295 – The Zeno chapel in the Santa Prassede church in Rome has a tetradic ceiling with four angels holding up Christ in a circle.

Charlemagne’s intentions to organize the intellectual heritage of the bygone Roman Empire and his Frankish-Celtic home ground, brought the tetradic way of thinking again to the surface. The historic presence of the European culture-as-a-whole could be found in the remnants of the Roman and Celtic cultural era. Architectural highlights, like the Zeno Chapel in Rome (ninth century), and the oratory of Germigny-des-Prés (805 AD; see fig. 201-202) carried the tetradic message into the ages to come.

The ceiling of medieval churches and chapels is often covered with geo-metric patterns, following a symmetrical order. A porch in the Cathedral of Wells, for instance, is a representative example of the creative possibilities offered in this field of architecture (fig. 296). The three-, four-, five-, six- and eight divisions are present in this simple and effective design. The cathedral of Wells – characterized by its curious inner supporting structure, which became necessary when the four massive piers started to sink after the construction of the central tower– was built in the early thirteenth century (dedicated in 1239).

Beautiful vault structures can be found in the adjoining Chapter House. This latter building dated from around 1250 and was completed under Dean John Godely (1305 – 1333). Basic rib vaulting is visible in the undercroft and so-called tierceron (from tierce, third) and lierne (lier, to bind) vaults mark the upper part of the polygonal building. The tierceron vaults are exploited here to the full with thirty-two ribs springing from the central column. Other examples can be seen in the cathedrals of Lincoln, Ely and Exeter.

This type of ceilings is part of the Geometric Style, which took over from the Early English Gothic Style in the middle of the thirteenth century (1240). The attention to geometry was widened in the Decorated Period Style, which lasted from 1275 to around 1360, when it was superseded by the Perpendicular Style. These latter three names were given by Thomas Rickman in his book ‘Attempt to Discriminate the Style of Architecture in England’, first published between 1812 and 1815.

Fig. 296 – Top: This drawing of a porch in the Wells Cathedral (England) gives an impression of the intricate nature of the ceiling. The four-, six- and eight divisions are dominant, but some of the bosses are centering points for a three- and five divisions. The triple bundle of ribs springing from the corners are called tiercerons (from tierce, third). The short linking ribs are termed lierne (from the French lier, to bind). Below: Another part of the ceiling of Wells Cathedral (photo: Marten Kuilman; 2009).

The vaults in the Grossen Rempter (refectory, dining room) in the castle of Marienburg (East Prussia, Germany) date from around 1320. They expose the creative powers of a multidivision pattern (fig. 297). The same patterns were copied in the salle des chevaliers of the castle of Stolzenfels, situated at the river Rhine, south of Koblenz. The construction of this latter castle consisted of various stages. The early beginnings in 1250 were followed by enlargements around 1370. The destruction in the Palatinate war in 1689 mends a temporary setback. The redevelopment in the early nineteenth century was carried out by the well-known German architect Karl Friedrich Schinkel (1781 – 1841) in a ‘Tudor style’.

Fig. 297 – The ceiling of the Grossen Rempter (refectory) in Marienburg (Germany), constructed around 1320, is an example of a well-executed and intricate geometric pattern.

The painting of the walls and ceiling of the church is a common feature. Even graffiti had its story to tell in English churches (PRITCHARD, 1967). Many sketches have references to the sign of the cross and are related with the four-fold (fig. 298).

The medieval graffiti on the walls of All Saints Church in Worlington (near Bury St. Edmunds) is related to the Third (?) Crusade – at least that was the suggestion by Pritchard in 1967. The heraldic sign of the King of Jerusalem (Godfrey of Bouillon) shows a pinned cross with four smaller crosses in the quadrants (fig. 298 – 1). Similar crosses can be found in the Holy Sepulchre church in Jerusalem, where pilgrims marked the walls with crosses throughout the ages.

The early English scholar and chronicler Matthew Paris stated, around 1250, that more than four crosses could be used: ‘scutum aureum crux alba cum multis parvis crucibis albis’ He pointed to the five gold (or yellow) crosses on a silver (or white) field. This setting is an exception of the rule of tincture in heraldry. This regularity stated that metal (bright tincture) must never be placed upon metal and dark colors (tinctures) should not be used upon colors, because of the lost of contrast. The heraldic sign of the Kings of Jerusalem became the arms of the Knights Templar, founded in 1118 to maintain a free passage for the pilgrims to the Holy Land.

The St. Margaret (of Antioch) Church in Cowlinge is situated in West Suffolk (England) close to the Cambridgeshire border. The church is the result of many alterations over the years and is ‘the kind of building a DIY enthusiast might put together if he wanted to make a late medieval one in his back garden’ (Simon Knott). The wall paintings (St. Michael balancing a set of scales) are probably created at the same time (fourteenth century) or slightly later as the octagonal piers. The theme (doom) reflects an opposi-tional frame of mind. The wealth of graffiti on the piers, including the Celtic knot work (fig. 298 – 2) is of various (later) periods. The brick tower of the church is added in the eighteenth century and contains a ring of five bells.

Fig. 298 – Medieval graffiti in some English churches is often associated with the sign of the cross.1. Heraldic sign of the king of Jerusalem in the All-Saints Church in Worlington (Suffolk, England); 2. Celtic knot work on the pillars of the St. Margareth Church in Cowlinge (Suffolk, England); 3. A swastika pelta in the St. Martin Church in Little Waltham (Essex, England); 4. A drawing in the same church (St. Martin); 5. A bend swastika with a cross made of points (St. Martin, Little Waltham); 6. Salomon’s knot in the St. John Church in Duxford (England).

The font with the octagonal bowl is a further point of interest in the church in Cowlinge (Suffolk, England). The bowl is situated at the west end near the entrance and dates from about 1400. A similar font can be found in the St Nicholas church in Denston, some five kilometers to the east. Here the eight panels depict the ‘seven sacraments’ with the crucifixion added to it (to make up eight): Matrimony (N); Baptism (NE); Ordination (E), Penance (SE); Mass (S); Last Rites (SW); Crucifixion and Confirmation (NW). The font of the church in Denston was dated between 1450 and 1485 and indicates a continuous attention to the symbolism of the octagonal even in times when oppositional thinking became the predominant state of mind.

The St. Martin Church at Little Waltham (Essex, England) hosts a number of interesting inscriptions (fig. 298 – 3/5) with the cross symbolism as their mutual inspiration. The burial index in the church in Little Waltham runs from 1538 – 1875, but the church itself – and possibly some of the graffiti – is older. One of the inscriptions (298 – 4) resembles the sand drawings of the inhabitants of Vao, an island in the Malekula archipelago in the small nation of Vanuatu, formerly known as the New Hebrides. The archipelago is situated between New Caledonia and the Fiji Islands.

The anthropologist Bernard DEACON (1934) published a collection of geometrical drawings in the Royal Journal of Anthropology of Great Britain and Ireland (fig. 299). This article is still a monumental publication on the topic of doodling and the inner human need to create signs (in a more organized way than graffiti). A correspondent of Deacon (miss Hardacre) told him that the lines were drawn ‘as pastime, chiefly by the young people when sitting about with nothing particular to do. They are traced with the finger in a nice moist patch of sand, well smoothed over, or in the dust, and are drawn on the frame work without removing the finger from the ground.’

Fig. 299 – This geometrical drawing from the island of Malekula (Vanuatu) reveals the universal human need to produce (tetradic) signs to visualize something else. The numbers indicate the way in which a levwaa (stump of a banana) was drawn as a continuous line in the sand.

The parish church of St. John’s in Duxford (10 km south of Cambridge, England) is a small gem of the twelfth century (nave, chancel and central tower). The northern chapel was added to the chancel around 1350 and the nave has a north aisle of around 1450. The wall paintings depict the crucifixion of Christ, St. Agatha’s martyrdom and some horrid scenes. The graffiti of Salomon’s knot (swastika pelta) (fig. 298 – 6) originate from the Middle East area. Doro LEVI (1947) described peltae from the ruins in the city of Antioch in his book on the mosaic pavements of that city. The St. John’s church was declared redundant in 1976 and is now only used for burial services. Most visitors prefer to go to the nearby Imperial War Museum, which records in such a vivid way the historical outcome of oppositional thinking.

An elaborate and tetradic orientated ceiling can be found in the southern portal of the Holy Trinity Church in the small village of Torbryan, six kilometers southwest of Newton Abbot in Devon (England) (fig. 300).

Fig. 300 – The roof of the southern portal of the Holy Trinity in Torbryan (Devon) is an example of a typical Perpendicular ceiling, which used the tetradic motif amidst the multiplicity of fan vaulting.

The church in Torbryan was constructed in one building campaign lasting from around 1450 to 1470 and belongs to the Perpendicular Style. Two arcades of arches run the length of the building. An octagonal stair turret at the south wall forms a dramatic architectural feature. The fan-vaulted ceiling has four small angels supporting the central ribs of each fan.

The Perpendicular Style (1330 to 1485) is the name given (by Thomas Rickman) to the final period of the English Gothic architecture and lasted from the late fourteenth to the middle of the sixteenth century. The name is derived from the predominating vertical lines and the prevailing angularity of the designs. The style is also known as Rectilinear or Late Gothic. The architectonic features developed from the Decorated style of the late thirteenth and early fourteenth century, but there is a gradual overlap between the periods. The latter part of the Perpendicular style period, after 1485, is sometimes classified as the Tudor style.

The small town of Burford (Oxfordshire, England), regarded by the tourist organizations as the ‘Gateway to the Cotswold’, is situated some thirty five kilometers west of Oxford and was once a thriving wool town. The High Street still has buildings dating back from the fifteenth century. The St John the Baptist parish church is a noteworthy piece of ecclesiastical architecture.

The original outlay of the church is cruciform and the central part of the tower and the west doorway are Norman. The rest of the building dates from around the fifteenth century. The southern porch of the church houses ‘one of the finest specimens of the Perpendicular Style of architecture in England’ (Gardner’s Directory ofOxfordshire, 1852) (fig. 301). The church played a role in one of the nasty stories of the English Civil War, when Cromwell and his men rounded up more than three hundred of the Levellers in May 1649. Three of the ringleaders were executed. The carvings and bullet holes of this event can still be seen today.

Fig. 301 – This part of the vault of the southern hall of the St John parish church in Burford (Oxfordshire) indicates a preference for the four-fold amidst the multitude of fan vaulting.

The development of English Gothic architecture can be seen from a quadralectic perspective and placed in a European context. The differences between England and the Continent might, despite a dissimilarity of terminology, less great in terms of division thinking. Only the degree of intensity and consciousness might have wavered at the various places of building.

The historic line, which is drawn by art historians, pointed to an ever more daring type of building in the years of the later Middle Ages. The European cultural history moved from the sturdy Romanesque Style (1000 – 1200) gradually into the Early Gothic (1150) and continued into the High Gothic (around 1200) with its ‘culmination’ in the Rayonnant Style of the Late Gothic Style (1250 – 1500) (see also fig 225).

This trend was, by-and-large mirrored in Great Britain by the Norman (1066 – 1180) – Early Gothic (1150) – Geometric/Decorated (1275 – 1360) and the Perpendicular style (or Late Gothic; 1360 – 1485) ending with the Tudor (after 1485). The latter style period can already be labeled as the English variety of the Renaissance, which swept all over Europe after 1450 after its initial start – in the beginning of the fourteenth century – in Italy.

Fig. 302 – The CF-graph of the Second (600 – 1200) and part of the Third Quadrant is given here for the European cultural presence. The ‘start’ (First Visibility, CF = 11) is allocated to the year 750 AD. A period of balance lasted from 750 – 900. The First Major Approach (FMA, CF = 6) was reached in the year 1050. The period between 1050 – 1350 is marked by increasing CF-values (from CF = 6 to CF = 13) leading towards the First Visibility Crisis (FVC) in 1350. Decreasing CF-values led to the Pivotal Point in 1500 (PP, CF = 10). The Romanesque and Gothic styles are given to compare their achievements in relation to a position on the CF-graph.

Nikolaus PEVSNER (1943/1961; p. 102) noted that ‘the Classic is only a moment in the history of a civilization’, and he distinguished a slowdown of the progressive forces in the Gothic style at the end of the twelfth century. There was a brief, new impulse shortly after the middle of the thirteenth century aiming for more light and space. The elevation became two-storeyed instead of three-storeyed and glazing replaced the last zones of dark masonry. France relaxed after 1275, according to Pevsner, but ‘the architecture of England between 1250 and 1350 was the most forward, the most important, and the most inspired in Europe’

The architectural shift in emphasis from the ‘round’ (Roman – Norman) to the ‘linear’ (Gothic) took place after 1050. The Communication Factor (CF), which expresses the theoretical shift value over the communication trajectory, had reached its lowest value (CF = 6) at this point (fig. 302). The full ‘depth’ of a communication between the European cultural history and an observer in the twenty-first century is experienced. This moment of the greatest intensio was followed by an upward movement towards the First Visibility Crisis (FVC) in 1350.

The ‘discovery of the linear’ was probably the greatest cultural event in the Gothic movement, leading to a landslide in the type of building. A degree of invisibility, which is always present in the circular, was abandoned in favor of the ‘straightness’ of the linear. The linear way offers a good prospect if a communication aims at power. The simplicity of the oppositional holds the key to a conscious identity in a communication.

The Rayonnant (on the continent) and the Decorated style (in Great Britain) searched in their ‘way up’ (higher CF-values) for the ultimate possibilities of the linear. The Perpendicular style, with its grid-like (rectangular) approach, found that boundary in the latter half of the fourteenth and fifteenth century. The grid (lattice), being a multiplication of the rectangular, is a dualistic continuation of a Third Quadrant graphic expression into the world of the Fourth Quadrant.

By the time Europe had reached its Pivotal Point (PP) in 1500 AD, there was little left of the creativity of the two-fold and imaginative people were trapped in the limitation of lower division thinking. The Renaissance – defined here as a movement looking back to the era of four-fold expression in the Roman Empire – was just the reflection of an escape, which Europe needed to break the constraints of an ever-increasing opposition.

However, this limiting state of mind brought out a spirit of expansion, which led to the great discoveries in geography and other fields of human activity. The invention of the printing press opened a new world of visibility. Knowledge became available to the masses on an unprecedented scale. The type of warfare changed dramatically with the large-scale introduction of gun-powder in the fifteenth century. Its deadly force left the less ‘advanced’ people in the world with no means to defend themselves. Limited thinking proved, initially and measured by its own standards, a profitable way of operating. Its ‘success story’ lasted for at least hundred and fifty years, until the ‘crisis’ around 1650 questioned its moral value.

A number of churches in Denmark, Sweden, Norway and Finland have medieval wall paintings depicting, among other iconographic subjects, the labyrinth. Most of them occur on the walls and vaults as frescos or graffiti. Thirty-two labyrinths at twenty-five locations have been located by John KRAFT and Jeff SAWARD (1991/2005). These authors suggest that the imagery of the predominantly ‘classical’ labyrinths was inspired by local folk tradition and not influenced by the church labyrinths in France, England and Italy. Stone labyrinths were widespread in Scandinavia and the theme was therefore, well known. They stated that ‘only at one location, at Grinstad church in Sweden, is it possible to trace an obvious, indisputable diffusion of continental influence to the Nordic church labyrinths’.

Most labyrinths in the Nordic churches date back to the fifteenth century and can be divided in four different geographical groups with their own characteristics. The largest (southern) group is in Denmark (twelve labyrinths) and southern Sweden (two labyrinths). All representations are fresco paintings and part of the official adornment of the church. The labyrinths at Tåning, Bryrup, Skørring and Nim in Jylland (Denmark) are of a similar type and seemed to be painted by the same artist.

The two Norwegian labyrinths (Seljord and Vestre Slidre) are not inside the church, but painted on the facade near the entrance. They might have had some protective purpose. The island of Gotland has five labyrinths, three in the form of graffiti (Hablingbo, Lye, Ganthem). Finland, or the eastern group, has eight labyrinths in four churches (Maaria, Korpo, Sibbo and Perna). They belong to a range of secular motifs like demons, mermaids, ships, dogs, etc. (fig. 303).

Fig. 303 – The labyrinth and other motifs on the ceiling of the Maaria Kyrka, in Turku, Southern Finland.

Ceilings reached new heights in the Renaissance. The term is – in a quadralectic interpretation – understood as a celebration of the visibility of the latter part of the Third and the following Fourth Quadrant of the Roman cultural period. The Romans looked at their Pivotal Point (125 BC; see fig. 88) desperately to the Greek. The European culture at its Pivotal Point (1500 AD; see fig. 267) needed the Roman culture to make advances in the artistic field. A quadralectic ‘law’ can be established by saying that:

A cultural entity at its historical Pivotal Point (the middle of the Third Quadrant) tend to look for its (architectural) inspiration to another, earlier culture, which had already progressed along this point and used the wider frame of mind of the Fourth Quadrant.

The Renaissance becomes, in this view, a retro-movement with the specific aim to make progress. Looking backwards is the only option for a two-fold mind if creativity is bogged down in the constraints of its own limitations. The initial experience of greater (visible) visibility will lose its attraction. This view to the past is, in itself, a dead end in the development of a culture-as-a-whole (CHASTEL, 1968). However, the action will pay out as the real nature of the search is discovered: if the yoke of dualism can be thrown off and a widened view makes deeper understanding possible. This historical event happened in the Roman culture in AD 125 (Emperor Hadrian) and in the European culture around 1800 (Emperor Napoleon).

The stucco in the Grand Salon of Villa Madama in Rome is one of many efforts to bring back the glory of the past. The Villa is one of the first Renaissance villas following the descriptions of similar buildings in Antiquity (such as the one given by the writer Pliny). The plan was drawn by Raphael, who left the completion to his followers Jules Romain (1499 – 1546) and Giovanni da Udine (1487 – 1564). They put together in the Grand Salon a classical fourfold scene with four medallions around a circular centerpiece with a quadriga (fig. 304). The Villa Madama is one of the most imitated villas of the Renaissance and its features appeared in many forms in the ages to come.

Fig. 304 – The ceiling of the Grand Salon in the Villa Madama in Rome was made after 1516 by Jules Romain (1499 – 1546) and Giovanni da Udine (1487 – 1564).

The tendency to follow the classical treatment of the ceiling continued from the sixteenth into the seventeenth century. The growing exuberance was later typified as the Baroque style. The power of numerology grew as a statement loaded with classical symbolism. However, the attention to all aspects of the four-division did not imply an increase in the four-fold way of thinking. The number ‘four’ was approached with all the vigor of a dualistic mind, leading to a numerological presence of tetradic features everywhere.

The occurrence of four-fold imagery and symbolism in the period following the Pivotal Point of the European cultural history (1500) is no proof for a growing dominance of the four-fold way of thinking. In fact, the opposite was true: the determination of lower division thinking to find the last examples of tetradic visibility became a pursuit in its own right.

The number of sculptures, paintings and illustrations with tetradic themes, like the seasons, the elements, the winds, quadriga’s, parts of the world, etc. is virtually endless in the period from the sixteenth century onwards. The architectural styles of this period reflect a growing influence of four-fold imagery – with the Italian architect Andrea Palladio (1508 – 1580) as a point of culmination. His Villa Capra (or Villa Rotunda) at Vicenza – built from 1567 to 1571 – is the ultimate tetradic building (see also fig. 505/506).

It is a sobering thought, in this context, that the writer of ‘I Quattro Libri dell’ Architettura.’ (1570; translated by TAVERNOR & SCHOFIELD, 2002 as ‘Palladio’s Four Books on Architecture’) lived in a time of predominantly oppositional thinking. It is fair to assume that his architectonic products and ideas, with their quadripartite notion, were the product of that very cultural spirit. This conclusion might seem weird for dualistic scientists, who are used to ‘count’ occurrences in order to rate their importance.

Any empirical existence, including the phenomenon of the four-fold, is part of a cycle of visibility, which is not ruled by measurable quantities, but by (two) points of recognition (POR). The choice of the latter is the result of an interaction by the observer and the observed over the whole field of communication. Recognition has to do with quantity, but is not ruled by its number. It is the intensity of its emotional response, translated in a degree of ‘visibility’, which makes the bond between an observer and the observed.

The key action in a quadralectic communication can be identified as a process of interconnecting the universe with a degree of relevance for our personal presence.

The ceilings by the French painter and art theorist Charles Le Brun (1619 – 1690) in Chateau de Vaux le Vicomte (around 1655) marked a decorative tribute to the classical four-fold. Le Brun was the dominant artist during the reign of Louis XIV, the Sun King. He went to Rome in 1642 and developed the theory of classicism. His aim was to create a codified system of art appreciation – an ambition to which the quadralectic approach feels much akin. The chateau, near Maincy in the Seine-et-Marne department of France, was built from 1658 to 1661, for Nicolas Fouquet (1615 – 1680), the finance minister in the government ran by Cardinal Mazarin. Fouquet’s success in politics and finances ended abruptly just three weeks after a famous feast at the chateau, when Louis XIV put him in jail. Fouquet learned the eternal lesson, that glory and defeat are two sides of the coin of power.

The late eighteenth and nineteenth century saw a renewed interest in the visual aspects of ceilings and their significance as bearers of a four-fold message. The Neoclassical and Romantic Movement found its inspiration in Roman palaces and villas. The decorated ceilings of the catacombs and in the ash-buried cities of Italy, like Pompeii and Herculaneum (discovered from 1750 onwards), added to the range of images used to create the time-honored symbolism of the four-fold and gave its makers a new perspective in due course.

The ceiling in the Ball Room in the building at 20 Portman Square, London, is a perfect example of this creative process in motion (fig. 305). It was created around 1780 and showed paintings by Angelica Kauffmann and plaster work in the style of Robert Adam (1728 – 1792). Robert Adam was a well-known architect, who was responsible for much of the Georgian development in London at the end of the eighteenth century. London had, by that time, surpassed Amsterdam as the wealthiest city in Europe. Many buildings and garden squares in the western suburbs of the city reflect Adam’s interpretation of neoclassicism. Syon House (1772) in west London, owned by the Duke of Northumberland, is one of his masterpieces.

Fig. 305 – Home House at 20 Portman Square Marylebone, London has this ceiling in the Ball Room. It is a specimen of a tetradic approach along the classical lines, with a circular and square element in a geometric setting.

A further investigation into the occurrence of tetradic structures and imagery on ceilings would probably reveal a continuous attention to this particular type of division. Future researchers will have to distinguish the importance of this type of division in its historic setting and try to relate its presence to a prevailing way of thinking. A synopsis of the mental structure should include the whole phase of visibility of a particular cultural complex as defined by the observer. The possibility of a various forms of division thinking at the same time must be kept in mind. The cryptic adagio that the actual occurrence of tetradic features does not necessarily point to a quadralectic frame of mind and vice versa is the hallmark of a modern way of research.

A modern description of architectural history – with its emphasis on finding the essence of visibility – puts strain on a researcher-to-be. The intellectual apparatus assumes full knowledge and understanding of the quadralectic way of thinking. The psychological tools of intuition (I), preliminary ideas (II), empirical knowledge (III) and interpretation (IV) should be finely tuned together and applied at the same time.

The inventor, architect and engineer Richard Buckminster Fuller (1895 – 1983) gave a substantial addition to the subject of roofs. His invention of a type of polyhedral framework became known as the geodesic dome. This roof construction covers a large space without internal support. The practical application of the dome was launched in the 1957, some thirty-five years after the first icosahedron concrete dome was made by engineers of the Carl Zeiss factory in Jena (Germany).

Fuller’s primary motto was to do more with less. The geodesic dome of the American Pavilion at the Expo 1967, now known as the Biosphère, on Île Saint-Hélène in Montreal, was the architectural crown on his ‘synergetic’ thinking. The geosphere of the Spaceship Earth Pavilion at Disney’s Epcot in Florida was another. His domes have since conquered the world and are still built for many purposes.

Buckminster Fuller’s dome experiment was based on the assumption, that the tetrahedron, octahedron and the icosahedron were the most stable geometric configurations in nature. The closest packing of four spheres results – after connecting their center points – into a tetrahedron, with four faces and four vertexes (fig. 306).

Fig. 306 – The American designer, architect and visionary Richard Buck-minster Fuller (1895 – 1983) proposed the tetrahedron as the most basic building stone in nature.

The stable packing of four spheres of equal diameter next to each other results in a perfect triangular configuration forming four angles of sixty degrees.

Buckminster Fuller went on to develop his sometimes controversial ideas, which were, nevertheless, practical and workable. He proposed the concept of ‘Spaceship Earth’, a systematic world view in which the planet was seen as a unity, traveling through the universe. Overspecialization was considered to be the root of the problems in the world at the end of the 1970’s (Manual for Spaceship Earth, 1969). He explored the principles of energy and material efficiency in the fields of architecture, engineering and design and can be regarded as one of the first initiators of global thinking. Mankind was seen as a passenger in a cosmic spaceship with the only luggage being energy and information.

A ‘Comprehensive Anticipatory Design Science’ could solve, in Fuller’s view, the problems of humanity. In the end ‘there is no energy crisis, only a crisis of ignorance’. His two-volume book on ‘Synergetics, explorations in the geometry of thinking’ (FULLER, 1975/79) is a mind-boggling exercise in creative thinking. The definition of synergy is given as a phenomenon in which the behavior of whole systems cannot be predicted by the behavior of their parts taken separately (101.00). Or, put in another way: ‘When individuals join in a cooperative venture, the power generated far exceeds what they could have accomplished acting individually’. This statement has a holistic character, which is a familiar entity in the last quadrant of the quadralectic communication (IV, 3).

The historical position of Buckminster Fuller’s ideas – surfacing around the year 1950 – is important in the development of architectural history in both America and Europe. The American culture was at the borderline between the Second and Third Quadrant (1946; fig. 551). It entered after the Second World War in a period of greatest visibility and became a world power. ‘American’ imagination, distributed by American technology, spread over the world. Fuller’s legacy was part of this intellectual renewal. Kirby URNER (1998) suggested that ‘his philosophy may be a source of creative ferment in new Continental brews (i.e. in European schools of thought)’.

Europe’s cultural position was at a turning point in the Fourth Quadrant of its presence (see fig. 267). It had reached, for the second time, a point of maximum intensio (with a CF-value of 6). This experience was filled with all the drama of the accumulated knowledge in the last quadrant of a quadralectic communication. Europe-as-a-cultural entity had seen it all in 1950. The Second War World was the last chapter in a cultural domination, which had lasted for ages. Time had come for a younger generation to take over and change the guard. Europe stood by as Russia and America performed their charade of a ‘Cold War’.

Fuller’s investigations of the universe led him from a packing of four circular entities (spheres) to the tetrahedron. The pyramid consisted of lines, which connected the four middle points of the spheres. The genetic history of the pyramid can be reduced to the circle. This geometric figure is identified, in the quadralectic approach, as a representation of the First Quadrant (I). The circle embodies the unity of an undivided universe, with no beginning and end.

The pedigree of the cube follows a different line. The roots are traced back to the square. The limitation of space by two (pair of) crossing lines is positioned in the Third Quadrant (III). The square defines the part as a dualistic limitation of the universe. The classical understanding of the part must, in a modern understanding, be placed in a foursome of different meanings. The unity of the First Quadrant (I) changes, after division, to the muun (multi-unity) of the Second Quadrant (II). The part in the Third Quadrant (III) is the proper part, characterized by the unity of the visible and, finally, the basic limitation in the Fourth Quadrant is called a whole.

The decoration of floors covers a large subject to study. They are a two-dimensional pendant to the three dimensional efforts of the roofs. Again only some of the more outstanding examples will be treated, leaving an enormous field for future investigators. The story of floor architecture is a major subject in its own right, with all the richness, skills and sometimes confusion of tetradic symbolism.

Maybe the most significant contribution to the treatment of floors was given by the Greek and Roman mosaic builders. The art flourished as an integral part of (interior) design from historic times, but found its highlights between the first century BC and the eighth century AD. Earlier mosaic work is known from the Chaldeans of around 2500 BC. The ‘standard of Ur’, now in the British Museum in London, is a classical specimen.

The Roman ‘Alexander mosaic’, depicting the confrontation of the Macedonians and the Persians, dated from the first century BC. The mosaic was used as a wall decoration in the ‘House of the Faun´ in Pompeii and was excavated in 1831. The mosaic, executed in the four basic colors (black, white, red and yellow), is now in the National Museum of Naples.

The heroic theme of Alexander the Great is, to a certain extend, an exception, because most motifs were domestically orientated and reflect scenes and items of everyday life, mythology and nature. The recently discovered (November 2005) mosaic on a site of the Megiddo prison in Israel (near the biblical Armageddon Valley), show two fishes in a circle amidst eight squares with tetradic motifs (including Salomon’s knot) (fig. 307) and three Greek inscriptions.

Fig. 307 – Part of a recently discovered mosaic in a quadrangular room from a religious place near Megiddo (Israel). The floor of the room measures approximately nine by six meters and is covered with common tetradic motifs.

The Megiddo mosaic is part of what might have been one of the oldest churches in the Holy Land. It dated from the late third century according to Yotan Tepper. This is before Constantine legalized Christianity across the Roman Empire in the early fourth century (313 – 323 AD). Some scholars are still skeptical about these claims and do not support the archeological dating. Comparison with other fish-panel compositions – like Beth Guvrin, Hazor-Ashdod, Beth Shean in Israel and the Procopius Church in Gerasa, Jordan – would place them in the sixth century. However, a mosaic from Tunesia with fish and shellfish pouring out of a rush basket – symbolizing great abundance – dated from the middle of the third century AD. (now in the Sousse Museum). It indicates that the fish motif was already present at that time.

Apart from the presence of two fishes in the Megiddo mosaic – and the knowledge that the motif was used outside Christian symbolism – there is only scanty evidence that the mosaic is part of a church at all (SLOFTRA, 2006). One of the inscriptions is concerned with four ladies and reads ‘You must remember Primilla and Kyriake and Dorothea and also Chreste’. These names are not known from other sources.

Primilla (‘the first’) was a common Roman name and Kyriake a proper (Greek) Christian name, which was frequently used in Antiquity. Dorothea can refer to the Christian Saint Dorothy, who died as a martyr in the district of Cappadocia (South-eastern Turkey) in the year 313, but was also a common name. Finally, Chreste (‘the kind one’) points to a certain form of character, although a reference to the Greek name ‘Christos’ (Christ) and ‘Khrestos’ (the kind/good one) cannot be excluded.

It would be an interesting speculation, from a quadralectic point of view, to see the ‘Four Women’ in a symbolic way and make them the personifications of a four-division. If the ‘fishes’ lead to Christianity (ichtus being the first letters of the words Jesus Christ), why could the ‘Four Woman’ not lead to some sort of tetradic thinking?

Another inscription reads: ‘Akeptus, the devout, dedicated the table to the God Jesus Christ as a memorial’. The mentioning of a table rather than an altar is curious, while the words ‘the God Jesus Christ’ is different from the common denominator as ‘Jesus, son of God’. The relation between the (Roman) four-fold way of thinking and the newly-started Christian belief in terms of division-thinking is an unexplored field of attention. In particular the change from the general, Fourth Quadrant, ‘Goodness’ of early Christianity (symbolized by Chreste) to the specific, Third Quadrant, presence as an orthodox ‘belief’ (symbolized by Dorothea) warrant further investigation. The character of the Second Quadrant (Kyriake) and the First Quadrant (Primilla, the First) could be added in a quadralectic orientated approach.

Tetradic themes and motives are common in (Roman) mosaics. The repeating patterns of black-and-white add an element of the two-fold. The variation of these patterns is endless, but its devotion to P4-symmetry is a common factor. It is no exaggeration to state that the main, collective feature of Roman mosaics is the display of four-fold symmetrical motifs (fig. 308).

This attention can be expressed in the individual (black-and-white) patterns, but also in the subject of mosaic as-a-whole. The theme of the ‘Four Seasons’ is present as far apart as the Roman mosaics in Tunesia and Libya (Zliten, Tripolitania) to the one in Brading Roman Villa on the Isle of Wight (England). Quadrigas and chariot racing scenes are depicted on the mosaics of the Piazza Armerina (Sicily) and are excavated at a Roman villa at Bingen (Germany) (FRAZER (1964). Many more representations of the tetradic themes are found in the area of the former Roman Empire.

A mosaic with Neptune on a quadriga surrounded by the four seasons, dating from the second century AD, is now exhibited in the Bardo Museum in Tunis (Tunisia). In the museum of Sousse, also in Tunisia, is a curious mosaic on display called ‘The Triumph of Dionysos’, with four tigers pulling the chariot (200 – 210 AD). The Louvre Museum in Paris (France) features a Roman mosaic with Neptune – and Amphitrite – on a quadriga, derived from Koudiat Ati near Constantine (Algeria), dating from around 325 AD.

Fig. 308 – The mosaic floors in the Hospitalia (guest rooms) of the Hadrian’s Villa in Tivoli (Italy) indicate elaborate tetradic patterns. The floors are from the first building phase of the complex between 118 and 125 AD.

The use of repeating motifs in mosaics find a zenith in the basilica of Misis (Mopsuhestia), south east Turkey. The decorations were discovered in 1956 and were originally laid down in the third quarter of the fourth century AD (fig. 309). The patterns in Kilikia are closely related with similar expressions in Syria (Antioch). They can be situated – as far as style is concerned – between the late-Constantinian villa in Daphne (just south of Antioch, with The Seasons in Room #1) and the ‘House of the Amazonomachy’ (Hunting Amazons)in the Yakto villa complex, also in Daphne (BUDDE, 1969, part I, p. 85).

Fig. 309 – The (reconstructed) mosaics of Misis (Moptuhestia) in Kilikia, south eastern Turkey, show distinct tetradic features (after BUDDE, 1969). This selection is part of the so-called ‘Samson cycle’ in the northern nave of the basilica, dating from the third quarter of the fourth century. The mosaics have suffered a great deal, not only by the earthquake of 526 AD, but also by robbers, who expected hidden treasures.

The geographic region of south eastern Turkey was encountered earlier as one of the key areas of cruciform (Greek cross type) ground plans (Ch. 3.3.1.2.1.; fig. 192, area B). The architectural forms of the ‘Thousand and one churches’ of Bin Bir Kilisse area – dating from the fifth to the eleventh century – are a natural continuation of the tetradic spirit, which found expression in the floor decorations of the early basilicas such as in Misis (Moptuhestia). Ludwig Budde noted the difference between the Christian and profane motifs on the mosaics (divided in his Band I and II), but it seems that the former is completely embedded in the latter. Other examples of Christian inspired mosaics were found in the churches of Dag Pazari and Aya-Elaeusa. However, they do not reach the quality of the mosaics in the basilica in Misis-Mopsuhestia.

Many of the profane mosaics, with themes and figures derived from the Greek mythology, four seasons, animals and plants, are now in the mosaic museum of Antakya (Antioch ad Orontes). The large mosaic of Oceanus and Thetis with the personification of the calendar (months and seasons) (second century) is the undisputed masterpiece. The Orpheus mosaic in Adana (Turkey) measures 1,56 x 1,85 meters and features as its middle part the celebrated Thracian musician surrounded by animals.

Two tritons are pulling a wagon on the Neptune mosaic (1,25 x 1,73 m), also in Adana. The god Neptune is only recognizable by his trident. The theme of the sea god riding the sea was very popular in Roman times all around the Mediterranean. The original four-fold symbolism (of the horses as the representatives of the four directions of the world) was not always strictly administered to for reasons, which can only be guessed (either by ignorance or as a deliberate statement). The excellent mosaic in La Chebba (near Sfax, Tunisia) is another classical picture of Neptune as the victorious driver of a quadriga (with two assistants, making up a total of seven figures).

The geographical area of cultural and religious movement, which hinges in southeastern Turkey, warrant a further investigation into the expressions of the tetradic, both in the Roman-Hellenistic culture as in the renderings of early Christianity. The designs and topics of the mosaics found around the Mediterranean are never scrutinized for a link with a particular type of division thinking. Geometric patterns were taken as expressions of visual art in their own right, but nobody made a connection with the state of mind of the artists (and their patrons).

The heydays of mosaics as ornamental art are not isolated events of artistic treatment of a plurality, but fit into the overall picture of a cultural presence. The attention to activities, which uses multiplicity to reach for new prospective horizons, can be viewed in the light of quadralectic visibility and its prominence in the various quadrants. The successive steps of a general creative process can be broken down as follows:

1. The process starts (arbitrarily) in the Second Quadrant (II) with the limitation of the invisible into the visible by division. It is likely that the two-division is of prime importance to delimit a Part. However, higher division-thinking could be introduced – theoretically – early in the process, changing the scope of the communication-as-a-whole.

2. The fixed concentration on the Part is a Third Quadrant (III) affair. Gaining identity is the main concern.

3. The organization of fragmentary and multitudinous parts takes place in the Fourth Quadrant (IV) and, finally,

4. These activities lead to an effusive unity, either in a retrograde direction, as a visibility in the Third Quadrant (III) – the mosaic in situ – or as a progressive change into the invisibility of the First Quadrant (I) of a new quadralectic cycle – as a work of art with its own, unwritten message.

Therefore, any imaginative activity with tesserae (the construction elements of mosaics), but also with knots (in rugs and tapestry) and/or – in a modern setting – with information (searching the internet), can be approached and validated in the above-mentioned cyclic communication.

The Roman upsurge of mosaic work in the first three centuries AD, for instance, seem to be an indication of the cultural move from the Third into the Fourth Quadrant of the Roman cultural period, around 125 AD (see fig. 88). The architecture and mosaics of Hadrian’s Villa in Tivoli are the zenith of a four-fold consciousness expressed in stone.

The Christian continuation of the mosaic tradition – both as floor and ceiling decoration – should be placed in the European cultural period. It covers the First Quadrant of that period (1 – 600 AD; see fig. 267) into the Second Quadrant. This switch of an architectonic feature like the mosaics, from one cultural period (Roman) to another (European) is a peculiar move in the eyes of conventional historians. Manifestation is not a unique event. It can be part of different (cultural) visibility periods at the same time. Its value in a communication cycle depends on the boundaries, which are drawn by the observer in the first place.

The mosaics in the Hagia Sophia in Constantinople were instigated by Emperor Justinian in the middle of the fourth century. They are a forceful statement of the young Christian identity to qualify as a cultural presence-in-the-making. Early Christian (ceiling) mosaic reached a peak in the fifth century in such churches as the St. George at Thessalonica, the nave of S. Maria Maggiore and St. John Lateran in Rome and the mosaic work in various churches in Ravenna. The mosaics in the inner roof of the Mausoleum of Galla Placidia (450) are of the same period.

Mosaic pavements of Roman origin flourished during the Christian era in countries around the Mediterranean and further north in Germany, France and Britain. The Roman mosaics lost their prominence in the cultural decline after 500 AD. However, fine pavements were still present in the seventh century, like the ones in ancient Tyre. Pope Honorius I (625 to 638) decorated in that period the apse of the St. Agnes in Rome. Pope John IV (640 – 642) placed mosaics in the chapel of St. Venantius at St. John Lateran’s and at St. Stephen’s on the Coelian Hill.

The manufacturing of pavement mosaics encountered a dismal time from the eight to the eleventh century. Wall mosaics were placed during this period in various Roman churches and in the S. Margaretta and St. Marks in Venice, in the St. Ambrose church in Milan and the Sancta Sophia at Constantinople. The mosaic from the latter (Hagia Sophia) is from the ninth century. It shows the Virgin Mary in the center, with Justinian I to the left holding a model of the church and Constantine I to the right holding a model of the city of Constantinople. The sack of Constantinople in 1204 implied a decline of mosaic art. The Hagia Sophia was restored after 1261 with a new Deesis (Christ Pantocrator) in the south gallery.

The floors of the later medieval churches were worthy successors of the Roman mosaics. They were often decorated with geometric patterns built up by tiles or intarso work. The latter consists of laid-in stone, meaning that certain forms are cut out of stone (often marble) and filled with a contrasting type of stone. In particular the Cosmas family made their name in this decoration technique and lend their name to a specific art style of marble inlays. Many buildings were decorated with ‘Cosmaten work’ between the twelfth and fourteenth century, like the churches of Santa Maria Maggiore, San Clemente and the S. Maria Maggiore in Rome. The (former) floor of the abbey church of Monte Cassino (Italy), dating from 1070 is one of the earlier examples of this style (fig. 310).

Fig. 310 – The former cosmaten floor of the abbey church of Monte Cassino (Italy) gives a variety of patterns in four-fold symmetry. The technique of marble inlay (intarso) is different from the manufacturing of mosaics proper.

The abbey of Monte Cassino, situated some hundred and twenty kilometers southeast of Rome, is the cradle of the Benedictine Order. This order, with its specific Rules, influenced the course of Roman Christian belief in Europe for a long time to come. It represented the ‘Third Quadrant’ part of the mental spectrum, using the strictness and clarity of lower division thinking to reach their religious goal.

Saint Benedict fled to the place (Cassium) around the year 529 AD, which was destroyed by the Goths some thirty years earlier. Only the temple of Apollo was still standing. Benedict, in turn, finished the job and destroyed the altar and the image of Apollo and built a church dedicated to St John the Baptist and an oratory in honor of St. Martin of Tours. The abbey continued to flourish after the saint’s death, until the Lombards pillaged it in 580. Restoration work started in 718 and a new church over the tomb of St Benedict was consecrated in 748 by Pope Zachary. The abbey had great fame in Europe and represents – in hindsight – a marker of the beginning of the visible visibility (CF = 11) of the European cultural period (arbitrary chosen in 750 AD, see also fig. 267, 274 and 302).

Tiles are a more crude way of covering a floor in comparison to the time-consuming works of mosaics and intarso. The intention is the same: to provide a visual pleasing representation of geometric patterns (fig. 311).

Fig. 311 – These arrangements of medieval tiles – as given by H. Kier – are inspired by P4-symmetry.

Many pavements in churches in Europe applied tiles to convey an artistic message. The Cistercian cloister of Bebenhausen (Baden-Württemberg, Germany) gives a representative example (fig. 312). The religious complex was founded shortly after 1185 and consecrated in 1228. Most of the tiles, as described by Horst DUBOIS (1995) and Ursula SCHWITALLA (1998), consist of groups of four. A unit is called a ‘quarreau’ (by Violet-le-Duc in his ‘Dictionaire d’Architecture’, II, p. 274).

Fig. 312 – Some of the tile patterns from the floor of the Cistercian cloister in Bebenhausen (Germany). Four tiles together (known as a ‘quarreau’) form a new unit, which can be repeated endlessly.

Interesting medieval tiles floors can also be found at many places in England (fig. 313). The comprehensive book by Jennie STOPFORD (2005) deals with tiles from Northern England. The hey-day of commercial ceramic tile-making took place during the reign of Queen Victoria (1837 – 1901).

Fig. 313 – A heraldic pavement, probably of the fifteenth century, was unearthed around 1817 during the excavation of King Edward’s Chapel in the ruins of the Abbey Church, Shaftesbury (Dorset, England).

The Renaissance period in the European cultural history implied that floors, just like roofs and walls, became surfaces to express the splendor of the long-passed Roman prominence. The geometric patterning disclosed a sense of order. Italian artists and architects of the fifteenth century took advantage of the ‘last creative escape’ from the straitjacket of oppositional thinking by looking at forms of multiplicity. They recognized that same urge in the Roman cultural period in the first centuries AD.

The construction of a floor pattern in Piero della Francesca’s painting ‘The Flagellation of Christ’ (in the Palazzo Ducale in Urbino, Italy) is a highlight of focused attention to measurement and an exercise in perspective (fig. 314). The German-American art historian Rudolf Wittkower (1901 – 1971) called the attention to geometry in the painting the ‘rationalization and harmonization of space and figures’ (WITTKOWER & CARTER, 1953). He suggested that the measurements, governing the various parts in the painting, were derived from Pacioli’s book ‘De Divina Proportione’.

Fig. 314 – A reconstruction of the floor pattern, which is part of Piero della Francesca’s painting ‘The Flagellation of Christ’. The polygonal division of the circle – which is seen in the painting under a very low angle – is intended to convey a message of perfection.

Pacioli’s book – written in 1494, but properly printed (and illustrated by Leonardo da Vinci) in 1509 – can be regarded as a hallmark of geometrical consciousness. The challenge of ‘squaring the circle’ (quadratura) and the establishment of the value of π (pi) became in the hands of Luca Pacioli (1445 – 1514) a superhuman mission. He tried – in his book ‘Summa de Arithmetica Geometria Proportione e Proportionalita’ (1497) – to connect the ‘holy’ circle (and cyclicity) with the ‘earthly’ square (and oppositionality). This aim, born in the spirit of duality, could be interpreted as a comparison between the First (circle) and the Third (square) Quadrant (see for the quadralectic distribution of these graphic expressions fig. 242).

The Golden Section is another phenomenon closely related to geometric proportion, which caught the enthusiasm of the Renaissance artists for the very same reason as the quadratura. The Sectio Aurea was already known by Plato (c. 427 – 347 BC) and Euclid (365 – 360 BC). The Greek sculptor and mathematician Phidias (500 – 432 BC) might have used the design for the sculptures on the Parthenon. Furthermore, the Egyptians long before him are said to have used the π (pi) and Φ (phi, the Golden Number) in the building of the Great Pyramids.

The Golden Mean is found when a line AB is divided in two parts by a point C in such a way that the ratio of the whole line AB to the longer segment AC is equal to the ratio of the longer segment AC to the smaller segment CB. If the length of AB (x = 1.000) then the position of C is approximately 0.618x. The division can be written, in a more mathematical form, like this:

The intriguing thing of this division is the fact, that the value of Φ (phi) is an irrational number (1.61803398874989…) and can never be exactly expressed in a whole number. This phenomenon added to the ‘mystery’ of the Golden Mean in the same way as the number π (pi) – giving the relation between the diameter and the circumference of a circle – did in the quadratura. The name ‘Divine Proportion’ (‘De Divina Proportione’), as Luca Pacioli introduced it, linked this irrational element to the presence of a Divine Being. The following (quadralectic) statement should be kept in mind when dealing with the above-mentioned curiosities:

The Golden Section can be interpreted as the ‘ultimate two-division’ of a line and is therefore, a genetic product of lower division thinking.

Leonardo da Vinci, the illustrator of Pacioli’s book in 1509, was probably the first to name the ‘sectio aurea’, the Latin translation of the Golden Section. His famous illustration of the ‘Vitruvian Man’, depicting a man with four arms and four legs within a circle and a square, became a symbol of the idea of proportion in a Renaissance universe. The (failed) mission of the Renaissance artists to join the divine (I) and the human (III) into a unity is an indication, that two-fold thinking encounters its own ‘mysteries’.

The Golden Section returned in the limelight pursuant to the Romantic Movement at the beginning of the nineteenth century. The curiosity of the ultimate two-division was mentioned as the ‘Goldener Schnitt’ in a scientific book by Martin Ohm (1792 – 1872; ‘Die reine Elementar-Mathematik’, 1835). He was the brother of the well-known scientist Georg Simon Ohm (1759 – 1854), who noted that the current through most materials is directly proportional to the potential difference across the material.

The name Ф (phi) for the Golden Mean was given in the early twentieth century by the American mathematician Mark Barr. Phi is the first letter of the before-mentioned sculptor Phidias, the Greek letter F and the first letter of Leonardo Fibonacci. The latter was a twelfth century mathematician, who came across some particular numerical series, which had affinities with the Golden Mean. It is not sure, however, if he realized its connection to Φ. Adolf Zeisung, in his book ‘Neue Lehre von den Proportionen desmenschlichen Körpers’ (1854), gave the subject of the Golden Mean a boast. He pointed to the occurrence in nature of several phenomena, following the ‘Fibonacci’ series.

The architect Le Corbusier tried to use the Golden Mean to generate ideal architectural dimensions of a building, but found the results ‘miserable’ (OSTWALD, 2000). His invention of the Modulor (fig. 315) is a replacement of the Golden Mean and can be used as a yardstick for the proportions of any building, and/or any other industrial mass design. The (Renaissance) idea ‘that human dimensions’ are essential in architecture is true, but to prove this point by a static number or set of numbers (like the Golden Mean or Modulor) is typical for lower division-thinking.

Fig. 315 – The Modulor by Le Corbusier (1950) is a measuring instrument in which the human body is used as a departure point for (architectonic) dimensions. A person with a height of 183 cm reaches with stretched arms to 226 cm. A division of this maximum length along the lines of the Golden Mean gives the ‘blue series’ (226 – 140 – 86 – 53, etc.). The so-called ‘red series’, has the navel (113 cm) as its point of departure, and consists of the values 113, 70, 43, 27 cm.

The idea of modular rules of design was already tried out by the Ionian architects Pytheos (fourth century BC) and Hermogenes of Priene (late third century BC). The former is known from the temple of Athena Pallas at Priene and the Mausoleum at Hallicarnassus and the latter as the architect of the temple of Artemis Leukophryene at Magnesia in Lydia (Anatolia, Turkey). The remark by David JACOBSON (1986; p. 72) that ’inspiration for the mathematical ideas came largely from the eastern Aegean’ is of particular interest in the search to roots of the tetradic spirit.

The southern Turkish coastal areas and inlands became a geographical contact point between the influences of the ancient Greek culture – with a moderate interest in particular divisions – and the Persians – with their continued attention to oppositional (division) thinking. The ‘tetradic spirit’ was already present in the Ionian philosophers of the sixth century BC, like Thales, Anaximander and Anaximenes (see also p. 121 – 123). They took the ‘best of both worlds’ and created a framework of four-fold thinking, which is still incorporated in the quadralectic philosophy.

The magic of the ‘Golden Section’ as an esthetic form-ideal was thoroughly investigated by Albert van der SCHOOT (1998). His skeptical conclusions with regards to the actual application in architecture were repeated by Marcus FRINGS (2002). They concluded that the popularity of the ‘Golden Section’ was of relative recent date. The myth, born in the nineteenth century, had a persistent life up to the present day. Le Corbusier and Neufert found out that the design tool was not particular helpful either.

Pythagoras’ statement that ‘everything was number’ offered an opportunity to develop a sense of quantitative proportion. Plato suggested, in his book Politeia (509d), a four-term proportion, which had the following terms:

illusion

conviction (belief)

reasoning and

insight (understanding).

The communication was – at least on the outset – positioned in a tetradic environment. The theory of nature, as unfolded by Plato in his Timaeus, was born in this same type of division-thinking. The concept aimed at a harmony between proportions and the four elements: fire was related to the tetrahedron, earth to a cube, air to an octahedron and water to a regular icosahedron (see fig. 266).

The understanding of symmetry, which is akin to the topic of proportion, was in Classical times a matter of commensurability. It pointed to the possibility to compare different entities in a communication with the same measure. The word ‘symmetry’ is derived from the Greek word sun (meaning ‘with’ or ‘together’) and metron (‘measure’). The summetria acts as the communication factor (CF) in the quadralectic theory. It is a quanti-fication of interaction between the communication partners.

It might be a bold statement, but the value of the CF-factor could be the ultimate presentation of Pythagoras’ belief (as recorded by his followers) that everything is number. Every interaction in every conceivable com-munication can be expressed in a number, which gives a precise appraisal of the distance between the communication partners. This number (CF-value) is, in due course, an indication of the degree of visibility in its widest sense.

The four synonyms for aesthetic arrangements in general can be organized in a quadralectic context. They are the representatives of four different views, which originate in the quadrants:

SZAMBIEN (1986) determined that the word ‘symetrie’ (written with a single m, the double m was only introduced from the eighteenth century) entered the French language in the year 1529. Apparently, the notion of this distinct aesthetic arrangement was so far advanced in the European cultural history, that a ‘new’ word was necessary. In general, it can be said that the process of naming is important as an indicator for conscious visibility and name giving can be, reciprocally, regarded as building blocks for human thoughts (KEES, 1956).

A different understanding of symmetry emerged in the seventeenth century, when the European cultural history reached its point of maximum oppositional thinking (the Second Visibility Crisis, 1650) on the boundary of the third (III, 3) and fourth part (III, 4) of the Third Quadrant. The original sense of an unspecified ‘measuring together’ (in a proportion) acquired a specific, pragmatic meaning, drifting towards balance and symmetry.

‘Measuring together’ became a ‘measuring apart together’ and the emphasis of the communication was shifted towards the parts rather than their initial unity. Seventeenth-century writers like Hume, Hogarth and Burke envisaged the beauty of symmetry as a ‘battle’ between individual parts, in which only the fittest could win.

Four types of movements to obtain a repetition of motives are recognized in the modern world of symmetry (the letters visualize the various actions):

Translation b to b

Rotation b to q

Reflection b to d

Glide reflection b to p

These actions take place between limited parts, which are interchangeable with respect to the whole. The minimum part that can be repeated without gaps of overlaps to make a complete pattern is called a fundamental region (STEVENS, 1980). All symmetries can be applied. The primitive cell (LAUWERIER, 1988) only uses translations.

The genesis of a fundamental region is a four-phased affair. This notion was brilliantly expressed in Escher’s ‘Houtsnede I’ (Woodcut I) (ESCHER, 1958; 1959/1970). Birds come to life between two pairs of parallel lines after the following operation: 1. Draw a contour line between A and B, which is then repeated between C and D; 2. Draw another contour line between A and C, which is repeated between B and D; 3. The basic geometric form is changed into a recognizable form (bird) and 4. The form is repeated in a grid (fig. 316).

Fig. 316 – The genesis of the fundamental region as the expression of the minimum area that can be repeated without gaps or overlaps to make a complete pattern. This example is a simplification of the twelve stages, which were given by Maurits Escher (1898 – 1972) in his book ‘Regelmatige Vlakverdeling’ (1958) as ‘Houtsnede I’ (Woodcut I).

The subject of proportion, and its interpretation in the seventeenth century, will be further discussed in chapter 5.1.4 as part of the ‘obligation to order’. Ceilings and pavements (tiles) will contribute a great deal to the impression of a building and tetradic features in the design of buildings can indicate a type of division thinking.

Much work, from a modern quadralectic perspective, has to be done to treat these elements in an appropriate way. This (re)search should not be performed in a numerological spirit, doggedly looking for any occurrence of the fourfold. Instead the interior features, like ceilings and floors, should be recognized in the full context of division thinking. The treatment and decoration of the inner walls and floors of a building is part of a greater concept of the division of space-in-general. This architectural reality, which permeates all types of visibility, holds the key to the type of division thinking of the communication process. Floors and ceilings and their decorations offer in their endless variety of forms a prime opportunity to discover the state of mind of their creators.